Anticommuting matricies

I know that two matrices anticommute when AB = -BA
However, apparently for n x n matrices when n is odd, either A or B is not invertible. But when n is even, both can be invertible
Why is this the case?

I know that two matrices anticommute when AB = -BA
However, apparently for n x n matrices when n is odd, either A or B is not invertible. But when n is even, both can be invertible
Why is this the case?

Look at the determinants: . If AB=–BA, it follows that either n is even or det(AB)=0.